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Let $f(x)$ be a non-zero polynomial with integer coefficients. Anautomorphism $\varphi$ of a group $G$ is said to satisfy the elementary abelianidentity $f(x)$ if the linear transformation induced by $\varphi$ on everycharacteristic elementary abelian section $S$ of $G$ is annihilated by $f(x)$.We prove that if a finite (soluble) group $G$ admits a fixed-point-freeautomorphism $\varphi$ satisfying an elementary abelian identity $f(x)$, where$f(x)$ is a primitive polynomial, then the Fitting height of $G$ is bounded interms of $\operatorname{deg}(f(x))$. We also prove that if $f(x)$ is anynon-zero polynomial and $G$ is a $\sigma'$-group for a finite set of primes$\sigma=\sigma(f(x))$ depending only on $f(x)$, then the Fitting height of $G$is bounded in terms of the number $\operatorname{irr}(f(x))$ of irreduciblefactors in the decomposition of $f(x)$. These bounds for the Fitting height arestronger than the well-known bounds in terms of the composition length $\alpha(|\varphi|)$ of $\langle\varphi\rangle$ when $\operatorname{deg} (f(x))$ or$\operatorname{irr}(f(x))$ is small in comparison with $\alpha (|\varphi|)$.

Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity